WASHINGTON — FBI leaders announced Tuesday that they are launching an independent investigation into the policies and actions of two bureau task forces that reviewed e-mails from the alleged Fort Hood shooter in the months before the Nov. 5 massacre at the Army base.

The inquiry will be headed by Wil liam Webster, a former director of the FBI and the CIA in the 1980s. He will have free rein to probe whether there were lapses in sharing information about Army Maj. Nidal Malik Hasan within the FBI and between that agency and the military.

Hasan, a military psychiatrist, has been charged with murder and attempted murder in the deaths of 13 people and the wounding of nearly three dozen others at the Texas base.

The action by FBI Director Robert Mueller III is the first significant signal since the attack that the bureau is concerned about its own actions. The Defense Department already had launched such an inquiry.

Hasan exchanged as many as 18 e-mail messages with radical Yemeni-American cleric Anwar al-Aulaqi between December 2008 and May 2009. But a joint terrorism task-force analyst determined that the correspondence was innocent and in keeping with the doctor’s research into religious conflicts among some Muslims in the military.

Two of those messages were forwarded this year from an FBI office in San Diego to one in Washington, where Hasan had worked at the Walter Reed medical facility. But a later e-mail that federal sources described as more serious was not shared with the Washington agents, a government official said.

In the weeks since the attack, Army personnel have come forward to express what they said were their earlier misgivings about Hasan. None of this information, however, appeared in military personnel records that the FBI analysts consulted in their assessment of Hasan’s e-mail.

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